#219888

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(2,4)(6,9)(7,10)(11,12)(13,14)(15,19)(16,24,26,20)(17,21)(18,22)(23,28,25,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,18,17)(14,19,22,21)(16,23,26,25)(20,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(2,4)(5,6,8,9)(7,10,11,12)(13,17,18,15)(14,21,22,19)(16,25,26,23)(20,28,24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dei,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127084);